Atiyah-Patodi-Singer index theorem from axial anomaly

TL;DR

This paper presents a straightforward derivation of the Atiyah-Patodi-Singer index theorem using path integrals of massless Dirac fermions, linking boundary conditions to physical states and the eta-invariant to axial charge.

Contribution

It offers a simple, physical derivation of the APS index theorem by modifying Fujikawa's approach, connecting boundary conditions with quantum states.

Findings

Derivation of APS index theorem via path integral method

Identification of boundary conditions as physical states

Relation of eta-invariant to axial charge

Abstract

We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions. It is based on the Fujikawa's argument for the relation between the axial anomaly and the Atiyah-Singer index theorem, and only a minor modification of that argument is sufficient to show the APS index theorem. The key ingredient is the identification of the APS boundary condition and its generalization as physical state vectors in the Hilbert space of the massless fermion theory. The APS $\eta$-invariant appears as the axial charge of the physical states.